Find the adjoint of the matrix A . Then use the adjoint to find the inverse of A (if possible).

Find the adjoint of the matrix A . Then use the adjoint to...

Key Concepts

Matrix Inversion Using the Adjoint

The process of finding a matrix inverse using the adjoint involves first computing the determinant and verifying that it is nonzero, then forming the cofactor matrix, transposing it to obtain the adjugate, and finally multiplying by the reciprocal of the determinant. This method provides a systematic approach to inverting a matrix when other more efficient computational methods might not be applicable.

Adjugate (Adjoint) Matrix

The adjugate, or classical adjoint, of a matrix is the transpose of its cofactor matrix. This matrix plays a central role in the formula for the inverse of a matrix, as it allows one to compute the inverse by scaling the adjugate by the reciprocal of the determinant, provided that the determinant is nonzero.

Cofactor Matrix

The cofactor matrix is composed of cofactors associated with each element of the original matrix. Each cofactor is computed by taking the determinant of the submatrix created by removing the row and column of a particular element, and then applying a sign factor of (?1)^(i+j). These cofactors are key to finding the adjugate matrix of a given matrix.

Determinant

The determinant is a scalar value that can be computed from the elements of a square matrix and provides important information about the matrix. In the context of finding an inverse, a nonzero determinant indicates that the matrix is invertible. It also plays a crucial role in formulas such as A?¹ = (1/det(A)) adj(A), where det(A) scales the adjugate matrix to yield the inverse.

Transcript

00:01 Other students we are given a matrix and we need to find the adjoint of a and if the its inverse exists then we need to find the inverse also so first of all we are supposed to know that adjoint of a can be written in the form of cofactors just like that c11 c1c12 c12 another column second column can be written as c22 c2 2 2 and c2 3 3 1 okay c31 c3 2 and c3 3 3 3 it's all about the adjoint of a now what we need to do we just need to find us c ij formula is nothing but minus 1 to the power i plus j times of m i j okay m i j so basically what we can do we just need to find the c1 c1 term can be written as minus 1 to the power 1 plus 1 which will be positive okay so we can write it directly m1 1 1 now it will be obviously 2 times of minus 2 is nothing but minus 4 minus minus 1 times of 3 is nothing but minus 3 so basically minus minus minus plus and it will be 3 so minus 4 plus 3 is nothing but minus 1 now what we need to do c 1 2 c 1 2 can be written as minus of m 1 2 so basically minus of m 1 2 can be written as see how we will how can we write that we just need to calculate that term 1 times of minus 2 is nothing but minus 2 okay and minus first of all we will write minus 2 okay and minus minus 1 times of 3 is nothing but minus 3.

01:30 Okay, so basically minus minus plus, okay, and 3.

01:34 So 3 minus 2 is nothing but 1 and 1 of minus of 1 is nothing but minus 1.

01:39 Now we need to calculate c 1 3.

01:41 Okay, so basically it will be positive can be written as m13.

01:44 M13 can be written as can be solved as 1 times of minus 1 is nothing but minus 1.

01:49 Okay, and minus of minus 1 times of 2 is nothing but minus 2.

01:54 So basically we can add minus 2 minus minus plus it will be 2.

01:57 So 2 minus 1 is nothing but 1 now we need to calculate c21 okay c2 2 2 and c2 3 as well so basically it will be negative because 2 plus 1 is 3 which is odd number so basically minus of m 2 1 it is m 2 1 it is nothing but minus of m 2 3 okay now we need to calculate all of those terms how we will calculate c minus of m 2 1 how we will calculate m 2 1 we need to skip that particular column in there through we will get 1 times of minus 2 nothing but minus 2 okay minus 1 times of 1 is nothing but minus 1 so basically minus minus minus plus and it will be 1 like that so minus 2 plus 1 is nothing but plus 1 okay and m2 we can escape that column in that through we will get 1 times of sorry 0 times of minus 2 is nothing but 0 okay minus minus 1 times of 1 is nothing but minus 1 so basically minus minus plus and really 1 so 1 plus 0 is nothing but 1 now we need to calculate c 2 3 okay so how we will calculate 2 3 okay so how we will calculate minus of m2 3 okay and how we will calculate m 2 3 just need to skip that particular column in that room we will get 1 sorry 0 times of minus 1 is nothing but 0 minus 1 okay so 0 minus 1 minus minus minus 1 okay so basically we will add 0 minus minus minus plus and 1 okay so 0 plus 1 is nothing but 1 minus 1 now we need to calculate the value of c sorry we need to calculate the last column which can be written as c31 c32 c33 okay how we will calculate those terms c31 c31 will be positive can be written as m 3 1 c32 will be negative can be written as minus of m 3 2 okay and here c33 will be positive can be written as m 3 now what we need to do we just need to find the m 3 1 we will find we can skip that particular column there through we will get 1 times of 3 is nothing but 3 minus 2 times of 1 is nothing but 2 so basically 3 minus 2 okay, we can write here 3 minus 2 can be written as 1 now minus of m 3 2 how we will calculate m 3 2 we just need to skip a skip that particular column in that row we will get 0 times of 3 is nothing but 0 minus 1 minus 1 so 0 minus 1.

04:13 So 0 minus 1.

04:13 Okay, we can write here 0 minus 1 so basically minus minus plus it can be written as 1 now what we need to do just need to calculate m 3 3 okay so m 3 3 3 how we will calculate we can write it as c...

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